Acceleration of Free Fall Experiment

Kinematic Foundation: The experiment relies on the SUVAT equation $s = ut + \frac{1}{2}at^2$. In this context, displacement $s$ is the height $h$, and acceleration $a$ is the gravity $g$.

Linearization Strategy: To account for a potential non-zero initial velocity ($u$) caused by delays in the release mechanism, the equation is rearranged into a linear form: $h = ut + \frac{1}{2}gt^2$. Multiplying by 2 and dividing by $t$ yields $\frac{2h}{t} = gt + 2u$.

Graphical Interpretation: When $\frac{2h}{t}$ is plotted on the y-axis against $t$ on the x-axis, the resulting line follows the form $y = mx + c$. The gradient of this line represents the acceleration due to gravity ($g$), and the y-intercept represents $2u$.

Apparatus Setup: A steel ball is held by an electromagnet above two light gates. Using a magnetic material allows for a controlled, instantaneous release when the current is cut.

Data Collection: The height ($h$) between the two light gates is measured using a metre rule. The time ($t$) taken for the ball to pass from the first gate to the second is recorded by a digital timer.

Variable Control: The experiment is repeated for multiple heights (typically 5 to 10 different values). For each height, multiple trials (at least 3) should be conducted to calculate an average time, reducing the impact of random errors.

Safety and Precision: A cushion is placed at the bottom to prevent damage to the ball or the floor. A glass tube may be used to guide the ball vertically, ensuring it passes through the center of the light gates.

Gradient Calculation: Always use a large triangle (covering at least half the line of best fit) to calculate the gradient. Ensure the units for the gradient are correctly identified as $\text{m s}^{-2}$.

Intercept Awareness: If the graph of $\frac{2h}{t}$ vs $t$ does not pass through the origin, do not force it. The intercept provides valuable information about the initial velocity $u$.

Unit Consistency: Ensure height is converted to meters ($m$) and time to seconds ($s$) before plotting. A common mistake is using centimeters, which results in a $g$ value of $981$ instead of $9.81$.

Uncertainty Analysis: Be prepared to calculate percentage uncertainties. The uncertainty in $t$ is often more significant than in $h$ because $t$ is squared in the base relationship ($h \propto t^2$).

Residue Magnetism: A common systematic error occurs when the electromagnet retains some magnetism after being switched off, causing a slight delay in the ball's release. This can make the recorded time longer than the actual fall time.

Parallax Error: When measuring the height $h$ with a metre rule, failing to view the scale at eye level can lead to consistent overestimation or underestimation of the distance.

Air Resistance: Students often assume $g$ will be exactly $9.81$. In practice, air resistance (drag) acts upwards, reducing the net acceleration and resulting in a calculated $g$ value slightly lower than the theoretical value.